Wake force excited by ultrarelativistic electron bunch in rectangular waveguide with periodic perturbed walls
А perturbation method for calculation of wakefields in a periodic waveguides with periodic perturbed walls is developed. The analytical formulas for a transverse component of an alternating wake force were derived. Розвинуто метод розрахунку кільватерних сил у слабогофрованих прямокутних хвилеводах....
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| Date: | 2008 |
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України
2008
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| Cite this: | Wake force excited by ultrarelativistic electron bunch in rectangular waveguide with periodic perturbed walls / A. Opanasenko // Вопросы атомной науки и техники. — 2008. — № 3. — С. 153-157. — Бібліогр.: 10 назв. — англ. |
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| author | Opanasenko, A. |
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| citation_txt | Wake force excited by ultrarelativistic electron bunch in rectangular waveguide with periodic perturbed walls / A. Opanasenko // Вопросы атомной науки и техники. — 2008. — № 3. — С. 153-157. — Бібліогр.: 10 назв. — англ. |
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| description | А perturbation method for calculation of wakefields in a periodic waveguides with periodic perturbed walls is developed. The analytical formulas for a transverse component of an alternating wake force were derived.
Розвинуто метод розрахунку кільватерних сил у слабогофрованих прямокутних хвилеводах. Виведено аналітичні формули для поперечних компонент знакозмінної кільватерної сили.
Развит метод расчета кильватерных сил в слабогофрированных прямоугольных волноводах. Выведены аналитические формулы для поперечных компонент знакопеременной кильватерной силы.
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| first_indexed | 2025-12-07T18:30:59Z |
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____________________________________________________________
PROBLEMS OF ATOMIC SCIENCE AND TECHNOLOGY. 2008. № 3.
Series: Nuclear Physics Investigations (49), p.153-157.
153
WAKE FORCE EXCITED BY ULTRARELATIVISTIC ELECTRON
BUNCH IN RECTANGULAR WAVEGUIDE WITH PERIODIC
PERTURBED WALLS
Anatoliy Opanasenko
NSC KIPT, Kharkiv, Ukraine
E-mail: opanasenko@kipt.kharkov.ua
А perturbation method for calculation of wakefields in a periodic waveguides with periodic perturbed walls is
developed. The analytical formulas for a transverse component of an alternating wake force were derived.
PACS: 41.60, 41.75.L, 41.75.H, 84.40.A
1. INTRODUCTION
The wakefield (WF) induced by a relativistic charge
particle bunch in a periodic corrugated waveguide and
the corresponding wake force can be expanded into Flo-
quet series in spatial harmonics. The spatial harmonics
synchronous with the bunch usually are of interest due
to their constant action on the particles that results in the
well-known beam loading and beam breakup instability
effects in the rf structures.
However, the alternating transverse wake force
which consists of the nonsynchronous harmonics can
give rise to undulating off–axis particles that should
result in no less important phenomena such as the wake-
field undulator radiation (WFUR) [1-5], and perhaps the
pondermotive focusing. A similar focusing, but in ex-
ternal fields, has been experimentally demonstrated in
Ref. [6,7]. Therefore there exists an interest in develop-
ing of methods of calculation of the alternating wake
forces induced by a relativistic charge bunch in the pe-
riodic rf structures.
Applying the method of field decomposition in the
eigen-modes in partially intersecting regions, the analyt-
ical formula of alternating wake force induced by a
bunch train of relativistic charged particles in the lowest
pass-band of the axial symmetric dick loaded wave-
guide has been derived in Ref. [8]. This results in the
possibility to perform under-estimations of WFUR pho-
ton fluxes from the periodic waveguide. Authors [9, 10]
have developed a perturbation method on calculating
wakefields, which takes into account all modes induced
by electron bunch in axially symmetrical waveguides
with periodic perturbed walls. However, the classical
round structures with sub-millimeter sizes, which need
for generation of hard WFUR [4,5], are difficult to con-
struct and be very expensive. So, in connection with
development of the micromechanic technology of fabri-
cation of planar sub-millimeter structures, the analytical
methods on calculating wakefields in the rectangular
periodic waveguides are relevant topic.
The goal of this paper consists in generalization of
the perturbation method [9,10] and expanding it on rec-
tangular waveguides with periodic perturbed walls.
2. PROBLEM STATEMENT
Let a bunch of N ultrarelativistic electrons moves
along a planar waveguide with weakly-corrugated me-
tallic both upper and lower surfaces. The wakefield and
the corresponding wake force, which acts on the bunch
electrons, have to be found.
In order to apply the method of Green’s functions
for arbitrary distribution of charge in the bunch, we will
find the wakefields excited by a point bunch, the current
density of which may be written as
( ) ( ) ( )0 0 vx y zj =0, j =0, j eN x x y y t zδ δ δ= − − − , (1)
where е is the electron charge, x, у are transverse coor-
dinates, z is a longitudinal coordinate, x0 and y0 are the
transverse coordinates of the point bunch, v is velocity
of the electrons, t is a time.
Consider both symmetrical and asymmetrical planar
periodic waveguides sketched in Figs.1, 2.
Fig.1. A rectangular waveguide symmetric with the
plane yz; x1,2=±b(z) are the surface contours, D is the
period
Fig.2. A rectangular waveguide asymmetric with the
plane yz; x1=b(z), x2=−b(z+D/2) are the surface contours
The periodic shape of the corrugated surface can be
represented by a Fourier series expansion
( )
2
0 01 , 0
pi z
D
p
p
b z b C e C
π
ε
∞
=−∞
⎡ ⎤
= + =⎢ ⎥
⎣ ⎦
∑ , (2)
where ε is a small parameter (0<ε <<1), b0 is the aver-
age half-distance between the upper and lower surfaces .
3. SOLUTION METHOD
3.1. EQUATION FOR WAKEFIELD
In periodic structures the electromagnetic fields
G=(E,H) excited by a charged particle moving in longi-
tudinal direction with velocity v may be represented by
the Fourier’s integral over frequencies ω and Floquet
series in spatial harmonics
( ) ( )
2
v
z pi t i z
D
p,t e e d
πω
ω ω
⎛ ⎞∞ ∞−⎜ ⎟
⎝ ⎠
⊥
= ∞−∞
⎧ ⎫⎪ ⎪= ⎨ ⎬
⎪ ⎪⎩ ⎭
∑∫ ,
p -
G r G r , (3)
where r=(x,y,z), r⊥=(x,y), Gω,p(r⊥) is the рth spatial har-
monic of the Fourier’s component.
154
Using the transformation (3) and expressing the
transversal field components through longitudinal, the
transversal components of the wake force spatial har-
monics may be found from the Maxwell equations as:
, , , ,
, , 2 2
2 2 v
v
p z p z
p x
p
E Hie p pF
D x D c y
ω ω
ω
π ω π
α γ
∂ ∂⎧ ⎫⎡ ⎤
= − −⎨ ⎬⎢ ⎥ ∂ ∂⎣ ⎦⎩ ⎭
,(4)
, , , ,
, , 2 2
2 2 v
v
p z p z
p y
p
E Hie p pF
D y D c x
ω ω
ω
π ω π
α γ
∂ ∂⎧ ⎫⎡ ⎤
= − +⎨ ⎬⎢ ⎥ ∂ ∂⎣ ⎦⎩ ⎭
, (5)
where we have introduced the following definitions:
( )2 2 2 221 1 v ; ; ;
vp p p
pc k c k k k
D
π ωγ ω α= − = = − ≡ − ,
с is the velocity of light.
The spatial harmonics of the longitudinal field com-
ponents satisfy the transformed wave equations
2
, , , , , ,2
4 2
v vp z p p z p z
i pE E j
Dω ω ω
π π ωα
γ⊥
⎛ ⎞
Δ + = −⎜ ⎟
⎝ ⎠
, (6)
2
, , , , 0p z p p zH Hω ωα⊥Δ + = , (7)
where 2 2 2 2x y⊥Δ = ∂ ∂ + ∂ ∂ ; jω,p,z is a current density
spatial harmonic of the point bunch Eq.(1)
( ) ( ), , 0 0 0,2p z p
eNj x x y yω δ δ δ
π
= − − . (8)
Here δ0,p is the Kronecher's symbol.
3.2. BOUNDARY CONDITIONS
The equations (6), (7) should be complemented with
the boundary conditions for conducting walls:
, , , , 0,
2 2z y
w wE x y z H x y z⎛ ⎞ ⎛ ⎞= ± = = ± =⎜ ⎟ ⎜ ⎟
⎝ ⎠ ⎝ ⎠
(9)
( ) ( )1,2 1,2, , , , 0.nE x x y z H x x y zτ = = = = (10)
For the symmetrical waveguide (Fig.3) a tangential
component of the electric field Eτ and a normal compo-
nent of the magnetic field Hn are expressed in terms of
the transverse components Ex, Ez і Hx, Hz, accordingly:
( ) ( )1,2 1,2, , sin , , cos 0x zE x y z E x y zα α± + = , (11)
( ) ( )1,2 1,2, , cos , , sin 0x zH x y z H x y zα α± − = , (12)
where ( )tg db z dzα = .
α
n1
τ
Ez
Ex
n2
x1
α
τ
Ez
Ex
b(z)
x2
αn1
τ
n2
x1
α
τ
Hz
Hx
b(z)
Hx
x2
α
n1
τ
Ez
Ex
n2
x1
α
τ
Ez
Ex
b(z)
x2
αn1
τ
n2
x1
α
τ
Hz
Hx
b(z)
Hx
x2
Fig.3. The boundary conditions for the symmetrical
waveguide
In case of the asymmetrical structure (Fig.4) the connec-
tion between the components Ez і Ex, Hx і Hz is given by:
( ) ( )1,2 1,2, , sin , , cos 0x zE x y z E x y zα α+ = , (13)
( ) ( )1,2 1,2, , cos , , sin 0x zH x y z H x y zα α± =m . (14)
α
n1
τ
Ez
Ex
α
n2
τ
Ez
Ex
x1
x2
b(z)
x1
x2
b(z)
α
τ
Hz
Hx
α
n2
τ
Hz
Hx
n1
α
n1
τ
Ez
Ex
α
n2
τ
Ez
Ex
x1
x2
b(z)
x1
x2
b(z)
α
τ
Hz
Hx
α
n2
τ
Hz
Hx
n1
Fig.4. The boundary conditions for the asymmetrical
waveguide
Expanding the fields on the surfaces x1,2 in terms of
a Taylor’s series close to the imaginary planes x=±b0
(see Figs.1,2) and taking into account Eq.(2), it is possi-
ble to get new boundary conditions:
− on the plane x=b0
( ) ( ) ( )
( ) ( )
( )
( ) ( )
, , 0
, , 0 0 2
, , 0
2
2
, , 02 2
0 2 2
2
, , 0 2
2
2
1
2
1 2
2
2 ,
q q z
p z p q
q q
q z
q
q q z
s p q s
q s q
q z
q
k E bp q
E b b C
D x
H bp q k
D y
k E bsb C C
D x
H bs k o
D x y
ω
ω
ω
ω
ω
π
ε
α
π
α
πε
α
π ε
α
∞
−
=−∞
∞ ∞
− −
=−∞ =−∞
⎡⎛ ⎞ ∂−
= − −⎢⎜ ⎟⎜ ⎟ ∂⎢⎝ ⎠⎣
⎤∂−
+ ⎥
∂ ⎥⎦
⎡⎛ ⎞ ∂
− −⎢⎜ ⎟⎜ ⎟ ∂⎢⎝ ⎠⎣
⎤∂
+ +⎥
∂ ∂ ⎥⎦
∑
∑ ∑
(15)
( ) ( )
( )
( ) ( ) ( ) ( )
, , 0 , , 0
2 2
0 , , 0
2
2 2
, , 0
, , 02
2
;
p z p z
p
p p q q z
qp q
q z q
q q z
H b E bk
x k y
b C E b
k
k x y
H b p q
k H b o
x D
ω ω
ω
ω
ω
α
ε
α
π α
ε
∞
−
=−∞
∂ ∂
= −
∂ ∂
⎡ ∂
− ⎢
∂ ∂⎢⎣
⎤∂ −
+ − +⎥
∂ ⎥⎦
∑
(16)
− on the plane x=-b0
( ) ( ) ( ) ( )
( ) ( ) ( )
( ) ( )
( ) ( ) ( )
, , 0
, , 0 0 2
, , 0
2
2
, , 02 2
0 2 2
2
, , 0 2
2
2
1
2
1 2
2
2 ,
q q z
p z p q
q q
q z
q
q q z
s p q s
q s q
q z
q
k E bp q
E b b C
D x
H bp q k
D y
k E bsb C C
D x
H bs k o
D x y
ω
ω
ω
ω
ω
π
ε
α
π
α
πε
α
π ε
α
±∞
−
=−∞
±
±∞ ∞
− −
=−∞ =−∞
±
⎡⎛ ⎞ ∂ −−
− = ± −⎢⎜ ⎟⎜ ⎟ ∂⎢⎝ ⎠⎣
⎤∂ −−
+ ⎥
∂ ⎥⎦
⎡⎛ ⎞ ∂ −
− −⎢⎜ ⎟⎜ ⎟ ∂⎢⎝ ⎠⎣
⎤∂ −
+ +⎥
∂ ∂ ⎥⎦
∑
∑ ∑
(17)
( ) ( )
( )
( ) ( ) ( ) ( )
, , 0 , , 0
2 2
0 , , 0
2
2 2
, , 0
, , 02
2
.
p z p z
p
p p q q z
qp q
q z q
q q z
H b E bk
x k y
b C E b
k
k x y
H b p q
k H b o
x D
ω ω
ω
ω
ω
α
ε
α
π α
ε
∞
−
=−∞
∂ − ∂ −
= −
∂ ∂
⎡ ∂ −
± ⎢
∂ ∂⎢⎣
⎤∂ − −
+ − − +⎥
∂ ⎥⎦
∑
(18)
Here and later on, the upper sign “+” corresponds to the
symmetrical structure, the lower sign “-“corresponds to
asymmetrical one.
3.3. ONE-DIMENSIONAL EQUATIONS
From the boundary conditions (9) it follows that the
longitudinal components of the fields may be written in
form of Fourier series with the period 2w
( ) ( ), , , , ,
1
, sin
2p z m p z
m
m wE x y E x y
wω ω
π∞
=
⎡ ⎤⎛ ⎞= +⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦
∑ , (19)
( ) ( ), , , , ,
1
, cos
2p z m p z
m
m wH x y H x y
wω ω
π∞
=
⎡ ⎤⎛ ⎞= +⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦
∑ . (20)
Thus Eqs.(6), (7) may be substituted by the one-
dimension ones:
( ) ( ) ( )
2
, , , 2
, , , , 0 0,2 2 ,m p z m
m p m p z p
E x iag E x x x
x
ω
ω
ωδ δ
γ
∂
+ =− −
∂
(21)
155
( ) ( )
2
, , , 2
, , , ,2 0m p z
m p m p z
H x
g H x
x
ω
ω
∂
+ =
∂
, (22)
02
4 sin
v 2m
eN m wa y
w w
π⎡ ⎤⎛ ⎞≡ +⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦
,
2
2 2
, .m p p
mg
w
πα ⎛ ⎞≡ − ⎜ ⎟
⎝ ⎠
For the boundary conditions (15)-(18) are the expan-
sion in terms of the small parameter ε , we will solve
Eqs.(21), (22) by a successive approximation method
expanding the fields in terms of the powers of ε:
( ) ( ) ( )
( ) ( ) ( )
0 1 2
, , , , , , , , , , , ,
0 1 2
, , , , , , , , , , , ,
...,
...
m p z m p z m p z m p z
m p z m p z m p z m p z
E E E E
H H H H
ω ω ω ω
ω ω ω ω
= + + +
= + + +
(23)
where ( ) ( )
, , , , , ,,n n n
m p z m p zE Hω ω ε∝ .
4. SOLUTION
4.1. ZERO ORDER WAKEFIELD
Leaving the zero order terms in both Eqs.(21)-(22)
and Eqs.(15)-(18), we can obtain
( ) ( ) ( ) ( ) ( )
02
0, ,0, 2
,0 , ,0, 02 2
m z m
m m z
E x iag E x x x
x
ω
ω δ
γ
∂
+ = − −
∂
, (24)
( ) ( )0
, ,0, 0 0m zE bω ± = , (25)
( ) ( ) ( ) ( )
02
0, ,0, 2
,0 , ,0,2 0m z
m m z
H x
g H x
x
ω
ω
∂
+ =
∂
, (26)
( ) ( )0
, ,0, 0 0m zH b
x
ω∂ ±
=
∂
. (27)
Omitting suitable calculations, recovering the time de-
pendence of the fields by the inverse Fourier transform
( ) ( ), ,2 Res nii
p p n
n
e d i eωτωτ
ω ωω π
∞
⊥ ⊥
−∞
⎡ ⎤= ⎣ ⎦∑∫F r F r (28)
(where τ=t−z/v), we obtain for the zero order only the
synchronous spatial harmonic of the wake field:
( ) ( )
( )
( ) ( )
( ) ( )
22
0
0
0,
10
v
2
0
1
0 0 0 0 0
0 0
0 0 0 0 0
0 0
8, , sin
2
sin 1
2
sin sin ,
2 2
sin sin ,
2 2
z
m
m s
w bs
s
eN m wE x y y
b w w
m wy e
w
s sx b x b x x b
b b
s sx b x b b x x
b b
π πγ τ
π πτ
π
π π
π π
∞
=
⎛ ⎞⎛ ⎞∞ − +⎜ ⎟⎜ ⎟
⎝ ⎠ ⎝ ⎠
=
⎡ ⎤⎛ ⎞= − +⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦
⎡ ⎤⎛ ⎞× + −⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦
⎧ ⎡ ⎤ ⎡ ⎤
+ − ≤ ≤⎢ ⎥ ⎢ ⎥
⎣ ⎦ ⎣ ⎦×⎨
⎡ ⎤ ⎡ ⎤
− + − ≤ ≤⎢ ⎥ ⎢ ⎥
⎣ ⎦ ⎣ ⎦
∑
∑
⎪
⎪
⎪
⎪
⎩
(29)
( ) ( )0
0, , , 0zH x y τ = . (30)
From Eq.(29) follows the well-known result [10], that
the zero order wakefield fall off exponentially in the
distance v⏐τ⏐~b0 /γ, that points out Coulomb’s nature of
this fields.
4.2. FIRST ORDER WAKE FORCE
Leaving the first order infinitesimals in Eqs. (21)-
(22), we have
( ) ( ) ( ) ( )
12
1, , , 2
, , , ,2 0m p z
m p m p z
E x
g E x
x
ω
ω
∂
+ =
∂
, (31)
( ) ( ) ( ) ( )
12
1, , , 2
, , , ,2 0m p z
m p m p z
H x
g H x
x
ω
ω
∂
+ =
∂
. (32)
These equations should be complemented with the
suitable boundary conditions:
− on the plane x=b0
( ) ( )
( ) ( )02
1 , ,0, 0
, , , 0 0
2 v1 m z
m p z p
E bpE b b C
D x
ω
ω
π γε
ω
∂⎛ ⎞
= − −⎜ ⎟ ∂⎝ ⎠
,(33)
( ) ( ) ( ) ( )
( ) ( )1 02
1, , , 0 , ,0, 0
, , , 0 0 2
0
m p z p m z
m p z p
p
H b E bk m E b b C
x k w x
ω ω
ω
απ ε
α
⎡ ⎤∂ ∂
= − +⎢ ⎥
∂ ∂⎢ ⎥⎣ ⎦
;
(34)
− on the plane x=-b0
( ) ( )
( ) ( )02
1 , ,0, 0
, , , 0 0
2 v1 m z
m p z p
E bpE b b C
D x
ω
ω
π γε
ω
∂ −⎛ ⎞
− = ± −⎜ ⎟ ∂⎝ ⎠
,(35)
( ) ( ) ( ) ( )
( ) ( )1 02
1, , , 0 , ,0, 0
, , , 0 0 2
0
m p z p m z
m p z p
p
H b E bk m E b b C
x k w x
ω ω
ω
απ ε
α
⎡ ⎤∂ − ∂ −
= − −⎢ ⎥
∂ ∂⎢ ⎥⎣ ⎦
m
.
(36)
Inserting the found E(1)
ω,p,z(x,y), H(1)
ω,p,z(x,y) into
Eqs.(19), (20), taking into account Eqs. (4), (5), and
applying the transform (28), we can obtain the time de-
pendence of the first order pth spatial harmonic of the
wake force transverse components:
( ) ( )
( ) ( ) , ,
2 2
1
,
1
,
0 0,
16
, sin
2
1
,
1
m p n
p
p x
m
n
i
m n
n n
e N pC m wF i y
wD w
X x eω τ
π ε πτ
δ
∞
⊥
=
∞
=
⎡ ⎤⎛ ⎞= +⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦
−
×
+
∑
∑
r
(37)
where ( ) ( ) ( )
( ) ( )
0
, 0 0 0
0 0
0 0 0
0
sin
2
cos
2 2sinh
cos
2
m n
m wy
w m nX x sh x b x b
mb w b
w
m nsh x b x b
w b
π
π π
π
π π
⎡ ⎤⎛ ⎞+⎜ ⎟⎢ ⎥ ⎧ ⎡ ⎤⎪ ⎡ ⎤⎝ ⎠⎣ ⎦= + +⎨ ⎢ ⎥⎢ ⎥⎛ ⎞ ⎣ ⎦⎪ ⎣ ⎦⎩⎜ ⎟
⎝ ⎠
⎫⎡ ⎤⎪⎡ ⎤± − − ⎬⎢ ⎥⎢ ⎥⎣ ⎦ ⎪⎣ ⎦⎭
( ) ( )
( ) ( ) , ,
2 2
1
, 2 2
10
,
1
2
, cos
2
1 m p n
p
p y
m
n i
m n
n
e N C D m wF i m y
b w p w
nY x eω τ
π ε πτ
γ
∞
⊥
=
∞
=
⎡ ⎤⎛ ⎞= +⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦
× −
∑
∑
r
,(38)
where
( ) ( ) ( )
( ) ( )
0
, 0 0 0
0 0
0 0 0
0
sin
2 sin
2 2
sin ,
2
m n
m wy
w m nY x sh x b x b
mb w bsh
w
m nsh x b x b
w b
π
π π
π
π π
⎡ ⎤⎛ ⎞+⎜ ⎟⎢ ⎥ ⎧ ⎡ ⎤⎪ ⎡ ⎤⎝ ⎠⎣ ⎦= + +⎨ ⎢ ⎥⎢ ⎥⎛ ⎞ ⎣ ⎦⎪ ⎣ ⎦⎩⎜ ⎟
⎝ ⎠
⎫⎡ ⎤⎪⎡ ⎤± − − ⎬⎢ ⎥⎢ ⎥⎣ ⎦ ⎪⎣ ⎦⎭
ωm,p,n are the frequencies of Čerenkov’s waves
22 2
, ,
0
v 2
4 2m p n
D m p n
p w D b
π π πω
π
⎡ ⎤⎛ ⎞⎛ ⎞ ⎛ ⎞⎢ ⎥= + + ⎜ ⎟⎜ ⎟ ⎜ ⎟
⎝ ⎠ ⎝ ⎠⎢ ⎥⎝ ⎠⎣ ⎦
, (39)
at γ→∞; ,0mg i m wπ= ± .
It should be noted that there is essential anisotropy
of transverse components of the alternative wake forces.
Thus, the у-component Fp,y ∼1/γ2 is strongly depressed
comparatively with Fp,x, because the electric у-
component of Lorentz’s force is compensated by the
magnetic one with an accuracy l/γ2.
4.3. SECOND ORDER WAKE FORCE
Electron energy losses connected with wakfield ex-
citation are defined the synchronous harmonic longitu-
dinal component of electric field which appear in the
156
series (23) for terms of the second order of smallness.
To find this field one needs to solve the equation
( ) ( ) ( ) ( )
22
2, ,0, 2
,0 , ,0,2 0m z
m m z
E x
g E x
x
ω
ω
∂
+ =
∂
(40)
with the boundary conditions on the plane x=b0
( ) ( )
( )
( )
( ) ( )
2
2
, ,0, 0 0
2 2 1
, , , , 0
2 2
,
2
1
1 2 v
m z q
q
m q m q z
m q
qE b b C
D
g E b
xq D g
ω
ω
πε
γ
π γ ω
∞
−
=−∞
⎛ ⎞= − ⎜ ⎟
⎝ ⎠
− − ∂
×
∂⎡ ⎤−⎣ ⎦
∑ (41)
and on the plane x=−b0
( ) ( )
( )
( )
( ) ( )
2
2
, ,0, 0 0
2 2 1
, , , , 0
2 2
,
2
1
.
1 2 v
m z q
q
m q m q z
m q
qE b b C
D
g E b
xq D g
ω
ω
πε
γ
π γ ω
∞
−
=−∞
⎛ ⎞− = ± ⎜ ⎟
⎝ ⎠
− − ∂ −
×
∂⎡ ⎤−⎣ ⎦
∑ (42)
Next, we find the time dependence of the synchronous
(zero) spatial harmonic of the longitudinal component
of the electrical field in the case γ→∞:
( ) ( )
( ) ( ) ( )
2 2 2
2 0
0,
1
2
, , , , ,
1 0 0,
8, , sin
v 2
1
2 cos ,
1
z
m
n
q m n m q n m q n
q n n
e N b m wF x y y
wD w
q C Z x
π ε πτ
ω ω τ
δ
∞
=
∞ ∞
= =
⎡ ⎤⎛ ⎞=− +⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦
−
×
+
∑
∑ ∑
(43)
where
( )
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
0
,
2 0
0 0 0 0 0
0 0 0 0 0
sin
2
2
1
1 .
m n
n
n
m wy
wZ x
mbsh
w
m m msh x b sh x b sh x b
w w w
m m msh x b sh x b sh x b
w w w
π
π
π π π
π π π
⎡ ⎤⎛ ⎞+⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦=
⎛ ⎞
⎜ ⎟
⎝ ⎠
⎧⎧ ⎫⎡ ⎤ ⎡ ⎤ ⎡ ⎤× − + ± − +⎨⎨ ⎬⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦ ⎣ ⎦⎩ ⎭⎩
⎫⎧ ⎫⎡ ⎤ ⎡ ⎤ ⎡ ⎤+ − − ± + −⎨ ⎬ ⎬⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦ ⎣ ⎦⎩ ⎭ ⎭
4.4. GAUSSIAN BUNCH
The wake forces derived above for a point charge
may be used as Green’s functions to find the force dis-
tribution within a bunch with arbitrary charge density as
( ) ( ) ( )2v, , , ,p p
0 S
d d
eN
τ τ ρ τ τ τ
⊥
∞
⊥ ⊥ ⊥ ⊥ ⊥′ ′ ′ ′ ′ ′= −∫ ∫∫F r r r F r r . (44)
We will consider a typical case when longitudinal
bunch size is considerably more than transversal, and
linear charge density is Gaussian distribution
( )
( )
( ) ( )
2
2
v
2
0 0, ,
2
z
z
eNx y e x x y y
τ
σρ τ δ δ
σ π
−
= − − , (45)
where σz is the root-mean-square bunch length.
The wake force components Eqs.(37), (38), (43), and
the density Eq.(45) are substituted into the definition
Eq.(44). As a result we get the wake force induced by
the Gaussian electron bunch:
( ) ( )
( ) ( ) ( )2
2
2 2
1
,
1
v
, ,, 2
0 0,
8
, , sin
2
1 v ,
1 v 2 2
z
p
p x
m
n
m p n zm n
n n
e NpC m wF x y i y
wD w
X x
e W i
τ
σ
π πτ ε
ω σ τ
δ σ
∞
=
∞ −
=
⎡ ⎤⎛ ⎞= +⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦
− ⎛ ⎞
× −⎜ ⎟+ ⎝ ⎠
∑
∑
(46)
( ) ( )
( ) ( )
( )2
2
2 2
1
, 2 2
10
v
, ,2
,
1
, , cos
2
v1 ,
v 2 2
z
p
p y
m
n m p n z
m n
n
e NDC m wF x y i m y
b w p w
nY x e W i
τ
σ
π πτ ε
γ
ω σ τ
σ
∞
=
∞ −
=
⎡ ⎤⎛ ⎞= +⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦
⎛ ⎞
× − −⎜ ⎟
⎝ ⎠
∑
∑
(47)
( ) ( )
( ) ( ) ( )2
2
2 2 22 2 0
0,
1 1
v
, ,, 2
, ,
0 0,
4, , sin 2
v 2
1 vRe ,
1 v 2 2
z
z q
m q
n
m q n zm n
m q n
n n
e Nb m wF x y y q C
wD w
Z x
e W i
τ
σ
π πτ ε
ω σ τω
δ σ
∞ ∞
= =
∞ −
=
⎡ ⎤⎛ ⎞=− +⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦
⎧ ⎫− ⎛ ⎞⎪ ⎪× −⎨ ⎬⎜ ⎟+ ⎪ ⎪⎝ ⎠⎩ ⎭
∑ ∑
∑
(48)
where ( ) ( )2
erfczW z e iz−= − [10].
5. EXAMPLE
The WF characteristics of the electron bunch with
eN=1 nC which moves along the waveguide with the
shape of the upper surface b(z)=b0[1+ε cos(2πz/D)]
(b0 = D = 0.3 mm, w = 10b0, ε = 0.1) are represented in
Figs.5-8. The structures both symmetrical and asymme-
trical are considered. Figs.5,7 and Figs.6,8 correspond
to the bunches both long σz=D and short σz=0.1 D, ac-
cordingly. The distributions of ⎪F1,x⎪ in x- direction
(Figs.5,6) point out surface wave character of the ex-
cited wakefield. A difference in the ⎪F1,x⎪ distributions
for the symmetrical and asymmetrical waveguides is
appreciable only close to the axis. F1,x equals to zero on
the axis of the symmetrical structure unlike the asym-
metric one.
Fig.5. The wake force transverse distribution for the
long bunch. 1-symmetrical, 2-asymmetrical waveguides
Fig.6. The wake force transverse distribution for the
short bunch. 1-symmetrical, 2-asymmetrical waveguides
In Figs.7,8 we presented the wake force distribution
along the bunch. These dependences have been built at
x0=0.75 b0, у0=0. It is interesting to note that in case of
the long bunch (σz= D, Fig.7) the wakefield localizes
within the bunch. The first half-bunch loses energy
while the second one adsorbs it fully without storing the
radiation into the waveguide after the bunch. In this case
the wakefield “spot” moves with the bunch velocity,
and the absolute value of the transverse component of
157
the alternating wake force F1,x reaches the maximum at
the maximum of the charge density where the longitu-
dinal component of the synchronous harmonic of the
electric field changes its sign. For the short bunch
(σz,=0.1 D, Fig.8) the WF distribution is classical, a
good part of the bunch loses energy to generate the wa-
kefield (the curve 1, F0,z), and the tail particles are re-
ceive both the maximal alternative transverse momen-
tum in the non-synchronous harmonics (the curve 2,
F1,x) and acceleration with a high gradient in the syn-
chronous harmonic (the curve 1, F0,z).
Fig.7. The WF distribution along the long bunch.
1- F0,z, 2-⎪F1,x⎪, the dash line is the charge distribution
Fig.8. The WF distribution along the short bunch.
1- F0,z, 2-⎪F1,x⎪, the dash line is the charge distribution
SUMMARY
The perturbation method for calculation of wake-
fields [9,10] have been generalized on the rectangular
waveguides with periodically perturbed walls with the
following main results:
In the symmetrical and asymmetrical structures the
wake forces are different only at the symmetry plane.
The transverse components of the alternating wake
force are strongly anisotropic.
For a long bunch the wakefield localizes within the
bunch. The transverse components of the alternating
wake force are maximal at the maximum of the charge
density.
For a short bunch the WF distribution is classical,
the maximum of the of the alternating wake force dislo-
cated toward the bunch tail.
The wakefields excited by an electron bunch passing
through the sub-millimeter planar periodic waveguide
may be used for both a high gradient acceleration and
generation of the wakefield undulator radiation.
REFERENCES
1. A. Opanasenko. Radiation by self-oscillating relati-
vistic charged particle moving along periodic struc-
ture // Proc. of MMET2002. Kyiv (Ukraine). 2002,
p.642-643.
2. A. Opanasenko. Radiation of charged particles in
self-wakefield // Proc. of RUPAC2002. Obninsk
(Russia). v.1, p.264-270.
3. A.Opanasenko. Characteristics of undulator-type
radiation emitted by bunch of charged particles in
wakefield // Problems of Atomic Science and Tech-
nology. 2004, №2, p.138-140.
4. A. Opanasenko. Conception of X-ray Source Based
on Compact Wakefield Undulator // Proc. of EPAC
2004. Lucerne (Switzerland).- P.2412-2414.
5. A. Opanasenko. Wakefield undulator for generating
X-rays // Proc. of RUPAC’04. Dubna (Russia).
2004, p.278-280.
6. Г.М. Иванов, В.И. Курилко, Л.А. Махненко и др.
СВЧ-фокусировка электронного пучка в уско-
ряющей структуре 4π⁄3 // Письма в ЖТФ. 1993,
т.19, №12, c.6-8.
7. J. Rosenzweig, L. Serafini. Transverse particle mo-
tion in radio-frequency linear accelerators // Phys.
Rev. E. v.59, №2, p.1599-1602.
8. A. Opanasenko. Analytical formulas for alternating
wake force of corrugated waveguides // Problems of
Atomic Science and Technology. 2006, №2, p.148-
150.
9. M. Chatard-Moulin, A. Papiernik. Electromagnetic
field accompanying a particle bunch in a periodic
waveguide with weakly perturbed wall // Nucl. In-
strum. and Meth. 1983, v.205 (1), p.27-36.
10. R.K. Cooper, S. Krinsky, L.P. Morton. Transverse
wake force in periodically varying waveguide // Par-
ticle Accelerators. 1982, v.12, p.1-12.
КИЛЬВАТЕРНАЯ СИЛА, ИНДУЦИРОВАННАЯ УЛЬТРАРЕЛЯТИВИСТСКИМ ЭЛЕКТРОННЫМ
СГУСТКОМ В ПРЯМОУГОЛЬНОМ СЛАБОГОФРИРОВАННОМ ВОЛНОВОДЕ
Анатолий Опанасенко
Развит метод расчета кильватерных сил в слабогофрированных прямоугольных волноводах. Выведены
аналитические формулы для поперечных компонент знакопеременной кильватерной силы.
КІЛЬВАТЕРНА СИЛА, ІНДУКОВАНА УЛЬТРАРЕЛЯТИВІСТСЬКИМ ЕЛЕКТРОННИМ
ЗГУСТКОМ У ПРЯМОКУТНОМУ СЛАБОГОФРОВАНОМУ ХВИЛЕВОДІ
Анатолій Опанасенко
Розвинуто метод розрахунку кільватерних сил у слабогофрованих прямокутних хвилеводах. Виведено
аналітичні формули для поперечних компонент знакозмінної кільватерної сили.
|
| id | nasplib_isofts_kiev_ua-123456789-111401 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1562-6016 |
| language | English |
| last_indexed | 2025-12-07T18:30:59Z |
| publishDate | 2008 |
| publisher | Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| record_format | dspace |
| spelling | Opanasenko, A. 2017-01-09T18:54:50Z 2017-01-09T18:54:50Z 2008 Wake force excited by ultrarelativistic electron bunch in rectangular waveguide with periodic perturbed walls / A. Opanasenko // Вопросы атомной науки и техники. — 2008. — № 3. — С. 153-157. — Бібліогр.: 10 назв. — англ. 1562-6016 PACS: 41.60, 41.75.L, 41.75.H, 84.40.A https://nasplib.isofts.kiev.ua/handle/123456789/111401 А perturbation method for calculation of wakefields in a periodic waveguides with periodic perturbed walls is developed. The analytical formulas for a transverse component of an alternating wake force were derived. Розвинуто метод розрахунку кільватерних сил у слабогофрованих прямокутних хвилеводах. Виведено аналітичні формули для поперечних компонент знакозмінної кільватерної сили. Развит метод расчета кильватерных сил в слабогофрированных прямоугольных волноводах. Выведены аналитические формулы для поперечных компонент знакопеременной кильватерной силы. en Національний науковий центр «Харківський фізико-технічний інститут» НАН України Вопросы атомной науки и техники Новые методы ускорения, сильноточные пучки Wake force excited by ultrarelativistic electron bunch in rectangular waveguide with periodic perturbed walls Кільватерна сила, індукована ультрарелятивістським електронним згустком у прямокутному слабогофрованому хвилеводі Кильватерная сила, индуцированная ультрарелятивистским электронным сгустком в прямоугольном слабогофрированном волноводе Article published earlier |
| spellingShingle | Wake force excited by ultrarelativistic electron bunch in rectangular waveguide with periodic perturbed walls Opanasenko, A. Новые методы ускорения, сильноточные пучки |
| title | Wake force excited by ultrarelativistic electron bunch in rectangular waveguide with periodic perturbed walls |
| title_alt | Кільватерна сила, індукована ультрарелятивістським електронним згустком у прямокутному слабогофрованому хвилеводі Кильватерная сила, индуцированная ультрарелятивистским электронным сгустком в прямоугольном слабогофрированном волноводе |
| title_full | Wake force excited by ultrarelativistic electron bunch in rectangular waveguide with periodic perturbed walls |
| title_fullStr | Wake force excited by ultrarelativistic electron bunch in rectangular waveguide with periodic perturbed walls |
| title_full_unstemmed | Wake force excited by ultrarelativistic electron bunch in rectangular waveguide with periodic perturbed walls |
| title_short | Wake force excited by ultrarelativistic electron bunch in rectangular waveguide with periodic perturbed walls |
| title_sort | wake force excited by ultrarelativistic electron bunch in rectangular waveguide with periodic perturbed walls |
| topic | Новые методы ускорения, сильноточные пучки |
| topic_facet | Новые методы ускорения, сильноточные пучки |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/111401 |
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